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5.1: Axioms of the Real Numbers
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In this chapter we will take a deep dive into structure of the real numbers by building up the multitude of properties you are familiar with by starting with a collection of fundamental axioms. Recall that an axiom is a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved.
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5.2: Standard Topology of the Real Line
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n this section, we will introduce the notions of open, closed, compact, and connected as they pertain to subsets of the real numbers. These properties form the underpinnings of a branch of mathematics called topology (derived from the Greek words tópos, meaning ‘place, location’, and ology, meaning ‘study of’). Topology, sometimes called “rubber sheet geometry," is concerned with properties of spaces that are invariant under any continuous deformation.